
The reciprocal lattice, however, is best looked at
as the Fourier
transform of the
regular lattice. We are showing this by constructing the Fourier transform of a
real crystal. 


It is easier to look at a real crystal (not just a lattice)
because otherwise you have to work with dfunctions. 


A real crystal has atoms. And atoms contain charge densities
r(r), or, if we start simple and
onedimensional, r(x). 


Now, r(x) must be
periodic in xdirection with the lattice constant
a: 


r(x + na) 
= 
r(x), 

n=0 ±1, ±2, ... 




We thus can expand r(x) into a
Fourier series, i.e. 


r(x) 
= 
S
n 
r_{n} · exp 
i · x· n· 2p
a 




The threedimensional case, in analogy, can be written as



r(r) 
= 
S
G 
r_{G}· exp 
(i · G· r) 




The vector G so far is just a
mathematical construct defining the "inverse" space needed for the
Fourier transform. 


However, since we can always substitute for any
r a vector r + T (T=
translation vector of the lattice), or written out, r +
n_{1}a_{1} +
n_{2}a_{2} +
n_{3}a_{3} with n_{i}=
integers and a_{i}= base vectors of the lattice
defining the crystal, the product r ·
G must not change its value if we substitute
r with r +
n_{1}a_{1} +
n_{2}a_{2} +
n_{3}a_{3}. 


This requires that G ·
T=2p · m with
m= integer. 

This is essentially a definition of the vectors
G that serve as the Fourier transforms of the vector
T, i.e. the lattice in space. These reciprocal lattice vectors, as they are called, can
be obtained from the base vectors defining the regular lattice in the following
way: 


If we write G in components
we obtain 


G 
= 
h · g_{1}+ k
· g_{2}+ l · g_{3}





With h, k, l= integers. 


The vectors g_{1},
g_{2}, and g_{3} are
then the unit vectors of the reciprocal
lattice. (yes – they are underlined, you
just don't see it with some fonts!) 

If we now form the inner product of
G · T, e.g., for simplicity, with
T= n_{1} ·
a_{1}, we obtain 


(h · g_{1}+
k· g_{2} + l ·
g_{3}) · (n_{1}·
a_{1}) 
= 
2p· m 




For an arbitrary n_{1} this
only holds if 


g_{1} ·
a_{1} 
= 
2p 







g_{2}·
a_{1} 
= 
g_{3}·
a_{1} 
= 0 



In general terms, we have 


g_{i}·
a_{j} 
= 
2p· d_{ij} 




With d_{ij}=Kronecker symbol,
defined by: d_{ij}=0 for ¹ and d_{ij}=1 for i=j. 

The above equation is satisfied with the following
definitions for the unit vectors of the reciprocal lattice: 


g_{1} 
= 
2 p· 
a_{2} × a_{3}
a_{1}· a_{2}·
a_{3} 




g_{2} 
= 
2 p· 
a_{3}× a_{1}
a_{1} · a_{2}·
a_{3} 




g_{3} 
= 
2 p· 
a_{1} ×
a_{2}
a_{1} · a_{2}·
a_{3} 

